Vector Algebra

Let $ABCD$be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $\left(\vec{AB}-\vec{BC}\right)+\left(\vec{AD}-\vec{DC}\right)=k\vec{FE}$, then $k$ is equal to

If the volume of parallelepiped formed by the vectors $\hat{i}+\lambda\,\hat{j}+\hat{k},\hat{j}+\lambda\,\hat{k}$ and $\lambda\,\hat{i}+\hat{k}$ is minimum, then $\lambda\,$ is equal to:

Let $\mathrm{ABC}$ be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{ck}$ and…

Let a vector $\alpha\,\hat{i}+\beta\,\hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area (in…

Let$x_{0}$ be the point of local maxima of…

Consider a$\Delta\,ABC$ where$A\left(1,3,2\right),B\left(-2,8,0\right)$ and$C\left(3,6,7\right)$. If the angle bisector of$\angle\,BAC$ meets the line $BC$at$D$, then the length of the projection of the vector$\vec{AD}$…

If the points $P$and $Q$are respectively the circumcenter and the orthocentre of a $\Delta\,ABC$, then $\vec{PA}+\vec{PB}+\vec{PC}$ is equal to _______

In a parallelogram $ABCD,\left|\vec{AB}\right|=a,\left|\vec{AD}\right|=b \left|\vec{AC}\right|=c$.$\vec{DB}\cdot\,\vec{AB}$has the value:

Let the three sides of a triangle $A B C$ be given by the vectors $2 \hat{i}-\hat{j}+\hat{k}, \quad \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$. Let $G$ be the centroid of the triangle $A B C$. Then…

If the vectors $\vec{\mathrm{AB}}=3\hat{i}+4\hat{k}$and$\vec{\mathrm{AC}}=5\hat{i}-2\hat{j}+4\hat{k}$are the sides of a triangle $ABC,$ then the length of the median through $A$ is:

In a triangle $ABC$,if $|\vec{BC}|=8,|\vec{CA}|=7,|\vec{AB}|=10$,then the projection of the vector $\vec{AB}$on $\vec{AC}$ is equal to :

A hall has a square floor of dimension $10m\times\,10m$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is $\cos\,^{-1}\frac{1}{5},$then the height of the hall (in meters) is: